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Illustrative Math
Grade 8
Lesson 9: Describing Large and Small Numbers Using Powers of 10
Let’s find out how to use powers of 10 to write large or small numbers.
Illustrative Math Unit 8.7, Lesson 9 (printable worksheets)
Lesson 9 Summary
Sometimes powers of 10 are helpful for expressing quantities, especially very large or very small quantities. For example, the United States Mint has made over
500,000,000,000
pennies. In order to understand this number, we have to count all the zeros. Since there are 11 of them, this means there are 500 billion pennies. Using powers of 10, we can write this as:
500 · 10^{9}
(five hundred times a billion), or even as:
5 · 10^{11}
The advantage to using powers of 10 to write a large number is that they help us see right away how large the number is by looking at the exponent.
The same is true for small quantities. For example, a single atom of carbon weighs about
0.0000000000000000000000199
grams. We can write this using powers of 10 as
199 · 10^{25}
or equivalently
1.99 · 10^{23}
Not only do powers of 10 make it easier to write this number, but they also help avoid errors since it would be very easy to write an extra zero or leave one out when writing out the decimal because there are so many to keep track of!
Lesson 9.1 Thousand Million Billion Trillion
 Match each expression with its corresponding value and word.
 For each of the numbers, think of something in the world that is described by that number.
Lesson 9.2 Baseten Representations Matching
 Match each expression to one or more diagrams that could represent it. For each match, explain what the value of a single small square would have to be.
 a. Write an expression to describe the baseten diagram if each small square represents 10^{4}. What is the value of this expression?
b. How does changing the value of the small square change the value of the expression? Explain or show your thinking.
c. Select at least two different powers of 10 for the small square, and write the corresponding expressions to describe the baseten diagram. What is the value of each of your expressions?
Lesson 9.3 Using Powers of 10 to Describe Large and Small Numbers
Your teacher will give you a card that tells you whether you are Partner A or B and gives you the information that is missing from your partner’s statements. Do not show your card to your partner.
Read each statement assigned to you, ask your partner for the missing information, and write the number your partner tells you.
Partner A’s statements:
 Around the world, about _________ pencils are made each year.
 The mass of a proton is _________ kilograms.
 The population of Russia is about _________ people.
 The diameter of a bacteria cell is about __________ meter.
Partner B’s statements:
 Light waves travel through space at a speed of _________ meters per second.
 The population of India is about _________ people.
 The wavelength of a gamma ray is __________ meters.
 The tardigrade (water bear) is ________ meters long.
Are you ready for more?
A “googol” is a name for a really big number: a 1 followed by 100 zeros.
 If you square a googol, how many zeros will the answer have? Show your reasoning.

Show Answer
We can write a googol as 10^{100}. Note that the exponent gives the number of zeros.
googol^{2} = (10^{100})^{2} = 10^{200}
which is 200 zeros
 If you raise a googol to the googol power, how many zeros will the answer have? Show your reasoning.

Show Answer
googol^{googol} = 10^{10010100} = 10^{100 · 10100}
which is 100 googol zeros.
Lesson 9 Practice Problems
 Match each number to its name.
 Write each expression as a multiple of a power of 10:
a. 42,300
b. 2,000
c. 9,200,000
d. Four thousand
e. 80 million
f. 32 billion
 Each statement contains a quantity. Rewrite each quantity using a power of 10.
a. There are about 37 trillion cells in an average human body.
b. The Milky Way contains about 300 billion stars.
c. A sharp knife is 23 millionths of a meter thick at its tip.
d. The wall of a certain cell in a plant is 4 nanometers thick. (A nanometer is one billionth of a meter.)
 A fully inflated basketball has a radius of 12 cm. Your basketball is only inflated halfway. How many more cubic centimeters of air does your ball need to fully inflate? Express your answer in terms of π. Then estimate how many cubic centimeters this is by using 3.14 to approximate π.
 Solve each of these equations. Explain or show your reasoning.
 Graph the line going through (6,1) with a slope of 2/3 and write its equation.
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